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The Link The Boltzmann Equation is the link S=k ln W S is entropy k is Boltzmann’s constant W is the number of configurations of the ensemble

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Statistical Mechanics and Dice If two dice are rolled and there is no distinction between them the outcome is called a combination A permutation is when the two dice are distinct from one another is different from A microstate is one arrangement of the dice or one permutation A configuration is the collection of all equivalent microstates and is defined as “W”

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Probability Probability (P): is a number between zero and one which tells the chance of a certain result or outcome occurring Probability is determined by the ratio of how many times an outcome can occur and how many possible outcomes there are.

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Rolling Dice Let’s compare the probability of rolling a 2 to the probability of rolling a 7 There are 36 possible outcomes when two dice are rolled. There is only one way to roll a 2 P = 1/36 = There are 6 possible ways to roll a 7 P = 6/36 = (graph on next slide)

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The Odds of Rolling A Number With Fair Dice

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Loaded Die There is no longer an equal likelihood for each of the six outcomes One of the outcomes is favored more then the others A weight function is used to describe how each outcome is favored

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Loaded Die (cont.) Example of a Weight Function w(n) = 1/n If one die was being rolled the probability of rolling a 6 would be determined by: The probability of rolling a one would be:

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Yes That’s Cheating When the loaded die is characterized by the weight function w(n)=1/n, the probability of rolling a 1 is six times greater than rolling a six /0.068=6 Depending on how the dice is loaded the probability of rolling a specific number increases (Graph on next slide)